Questions taken from Stackoverflow

This page lists questions about automata and other rational/regular expressions that were asked on Stackoverflow, and where Vcsn seems to be an appropriate tool to compute the answer.

Build a Regular Expression and Finite Automata

The set of all strings beginning with 101 and ending with 01010.

First, let's define our "context": we work with "labels are letters" (lal), on the alphabet $\{0, 1\}$. We don't use weights, or rather, we use the traditional Boolean weights: $\mathbb{B}$.

In [1]:
import vcsn
c = vcsn.context('lal_char(01), b')
c
Out[1]:
$\{0, 1\}\to\mathbb{B}$

Then, we build our expression using an unusual operator: $\mathsf{E} \& \mathsf{F}$ denotes the conjunction of expressions $\mathsf{E}$ and $\mathsf{F}$. In this case (unweighted/Boolean automata), it denotes exactly the intersection of languages.

In [2]:
e = c.expression('(101[01]*)&([01]*01010)')
e
Out[2]:
$1 \, 0 \, 1 \, \left(0 + 1\right)^{*} \& \left(0 + 1\right)^{*} \, 0 \, 1 \, 0 \, 1 \, 0$

We want to normalize this extended expression (it has conjunction and complement operators) into a basic expression. To this end, we first convert it to an automaton.

In [3]:
a = e.automaton()
a
Out[3]:
%3 I0 0 0 I0->0 F9 1 1 0->1 1 2 2 1->2 0 3 3 1->3 0 6 6 2->6 1 4 4 3->4 1 4->4 0, 1 5 5 4->5 0 5->6 1 7 7 6->7 0 8 8 7->8 1 9 9 8->9 0 9->F9

and then we convert this automaton into a basic expression:

In [4]:
a.expression()
Out[4]:
$1 \, \left(0 \, 1 + 0 \, 1 \, \left(0 + 1\right)^{*} \, 0 \, 1\right) \, 0 \, 1 \, 0$

Or, in ASCII:

In [5]:
print(a.expression())

1(01+01(0+1)*01)010

Regular expression to match text that doesn't contain a word?

I'd like to know if it's possible to match lines that don't contain a specific word (e.g. hede) using a regular expression?

First, let's define that alphabet we work on: from $a$ to $z$ for instance.

In [6]:
import vcsn
c = vcsn.context('lal_char(a-z), b')
c
Out[6]:
$\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z\}\to\mathbb{B}$

Then we define our expression, which is extended (it uses the complement operator), so to normalize it, we first convert it into automaton (with _expression_.automaton), from which we extract a basic expression (with _automaton_.expresion).

In [7]:
e = c.expression('(hede){c}')
e
Out[7]:
$\left(h \, e \, d \, e\right)^{c}$
In [8]:
a = e.automaton()
a
Out[8]:
%3 I0 0 0 I0->0 F0 F1 F2 F3 F4 0->F0 1 1 0->1 [^h] 2 2 0->2 h 1->F1 1->1 [^] 2->F2 2->1 [^e] 3 3 2->3 e 3->F3 3->1 [^d] 4 4 3->4 d 4->F4 4->1 [^e] 5 5 4->5 e 5->1 [^]
In [9]:
a.expression()
Out[9]:
$\varepsilon + h \, \left(\varepsilon + e \, \left(\varepsilon + d\right)\right) + \left([\hat{}h] + h \, \left([\hat{}e] + e \, \left([\hat{}d] + d \, \left([\hat{}e] + e \, [\hat{}]\right)\right)\right)\right) \, {[\hat{}]}^{*}$

Or, in ASCII (+ is usually denoted |; \e denotes the empty word; and [^] denotes any character, usually written .):

In [10]:
print(a.expression())

\e+h(\e+e(\e+d))+([^h]+h([^e]+e([^d]+d([^e]+e[^]))))[^]*

Convert finite state machine to regular expression

Is there a tool (or an algorithm) to convert a finite state machine into a regular expression?

Vcsn is tool for rational expressions and automata. See http://vcsn.lrde.epita.fr.

In [11]:
import vcsn

Build a random automaton $a$ of 4 states, labeled by letters to choose in ${a, b, c}$. Then "lift" it into an automaton $b$ labeled by expressions.

In [12]:
a = vcsn.context('lal(abc), b').random_automaton(4)
b = a.lift()
b
Out[12]:
%3 I0 0 0 I0->0 F1 2 2 0->2 ⟨c⟩ 1 1 1->F1 1->1 ⟨a⟩ 3 3 2->3 ⟨a+b⟩ 3->1 ⟨b⟩

Eliminate state 2, then 3, etc. The order does not matter for correction (any order gives a correct result), but the quality of the result does depend on this order.

In [13]:
b = b.eliminate_state(2)
b
Out[13]:
%3 I0 0 0 I0->0 F1 3 3 0->3 ⟨c(a+b)⟩ 1 1 1->F1 1->1 ⟨a⟩ 3->1 ⟨b⟩
In [14]:
b = b.eliminate_state(3); b
Out[14]:
%3 I0 0 0 I0->0 F1 1 1 0->1 ⟨c(a+b)b⟩ 1->F1 1->1 ⟨a⟩
In [15]:
b = b.eliminate_state(1); b
Out[15]:
%3 I0 0 0 I0->0 F0 0->F0 ⟨c(a+b)ba*⟩

Eliminate the last state, and read the answer.

In [16]:
b = b.eliminate_state(0); b
Out[16]:
%3 Ipost Fpre Ipost->Fpre ⟨c(a+b)ba*⟩

Alternatively, you can use the method automaton.expression to ask for the result.

In [17]:
a.expression()
Out[17]:
$c \, \left(a + b\right) \, b \, {a}^{*}$

You may see this algorithm run "interactively" using %demo.

In [18]:
a = vcsn.context('lal(abc), b').random(10)
b = a.lift()
%demo b eliminate_state
%3 I0 0 0 I0->0 F8 1 1 0->1 ⟨b⟩ 3 3 0->3 ⟨a+b+c⟩ 5 5 1->5 ⟨b⟩ 6 6 1->6 ⟨c⟩ 9 9 1->9 ⟨b+c⟩ 2 2 7 7 2->7 ⟨b+c⟩ 8 8 2->8 ⟨a+c⟩ 4 4 3->4 ⟨b⟩ 4->7 ⟨a+b+c⟩ 5->2 ⟨a+c⟩ 5->7 ⟨a+c⟩ 6->6 ⟨c⟩ 7->1 ⟨b⟩ 7->7 ⟨b+c⟩ 8->F8 8->4 ⟨a+c⟩ 9->0 ⟨b⟩ 9->9 ⟨a+b⟩

Constructing a Regular Expression from a Finite Automata

I'm trying to construct a regular expression from a Finite Automaton but found my self completely stuck with this one.

In [19]:
%%automaton a
$  -> q3
q1 -> q2 a
q1 -> q3 b
q2 -> q2 a, b
q3 -> q4 a
q3 -> q3 b
q4 -> q4 a
q4 -> q1 b
q3 -> $
q4 -> $
%3 I0 0 q3 I0->0 F0 F3 0->F0 0->0 b 3 q4 0->3 a 1 q1 1->0 b 2 q2 1->2 a 2->2 a, b 3->F3 3->1 b 3->3 a
In [20]:
a.expression()
Out[20]:
$\left(b + a \, {a}^{*} \, b \, b\right)^{*} \, \left(\varepsilon + a \, {a}^{*}\right)$